300 Mr. Knight on the Attraction of such Solids 
If r is finite, the above expression will terminate when m is 
a whole positive even number; and consequently the guiding 
curve will then be algebraic. But, if m be amongst the num- 
bers 5, 7, 9, 11, & c., we must use the other expression found 
in the lemma, and there arises, for the guiding curve, the 
transcendent equation 
*'<P (x, ™,y, ry) + Cry = 0. 
If m = i, the equation is 
-y z 
when m 
x arc (tang. 
3 . 
ry 
-f- Cry = 0 ; and, finally, 
-T7-7 x -rrPz % + rT7~ru arc (tang. = - 7== + Cry = 0 . 
In like manner, might be solved Prop. 31 and 34,, the force and 
density being as in Lemma 3, but this I leave to the reader. 
Prop. 38. 
The force being inversely as the mth power of the distance 
(where m is any whole positive number), and the density 
either uniform or any function of x and T t * the base of the 
infinitely long cylinder of greatest attraction has, for its 
equation, 
— ; ~ — 4- C^o; 
(**+/)* 
for it will appear from lemma 3, and its corollaries, that, 
whether m be odd or even (that is to say when it is any num- 
ber in the series 1, 2, 3, 4, $, &c.), the attraction of an infinite 
cylinder will be of the form 
* What this means with respect to a cylinder, is shewn at the end of the scholium 
to Prop. 33; and with respect to a solid of revolution in Prop. 33. 
